An internal combustion engine oil supply system wherein a common single suction port possessed by a pair of oil pumps sharing a pump casing is connected to an oil strainer and wherein relief valves are individually connected to discharge passages of the oil pumps has been known, as disclosed in Patent Document 1.    Patent Document 1—Japanese Patent No. 4467412
In the system disclosed in the Patent Document 1, the oil discharged from the relief valve of each of the oil pumps is returned to the suction side of the corresponding oil pump. In the case where the oil discharged from one of the pair of oil pumps is used as a lubricating oil supplied to to-be-lubricated portions of engine body parts whereas the oil discharged from the other oil pump is used for control of oil pressure in a transmission, however, a circulating circuit may be so configured that the oil discharged from the other oil pump is entirely returned to the suction side of the other oil pump through the relief valve at other times than the time of a gear shift. In this case, since the suction port is common to both the oil pumps, a portion of the oil to be sucked by the one oil pump may be sucked into the circulating circuit side. In such a situation, if a rapid acceleration or rapid deceleration of the vehicle is conducted, the oil surface of the oil in the engine body may be oscillated back and forth so that a suction port of an oil strainer may be exposed from the oil surface, resulting in that air is sucked into the oil strainer together with the oil. If the oil thus admixed with air is sucked into the circulating circuit in a large quantity, it may influence the control of oil pressure in the transmission. In order to prevent such a situation from occurring, therefore, it is necessary to take a countermeasure, such as enhancing the pump capacity of the one oil pump, which may lead to an increase in pump size.